An Improved Bound for Plane Covering Paths

TL;DR

This paper presents a new upper bound of approximately 6/7 times the number of points for the minimum segments needed in a plane covering path, improving previous bounds and providing an efficient algorithm.

Contribution

The authors establish a tighter upper bound for plane covering paths and introduce a constructive, efficient algorithm to compute such paths.

Findings

Proved that n/7 is an upper bound for plane covering paths.

Improved the previous bound of 21n/22.

Provided an O(n log n) algorithm for constructing the paths.

Abstract

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $\pi(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $\pi(n)$ segments. We prove that $\pi(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.